ADDITION AL NOTES. 51 



Mr. Aii-y (Tides ami Waves, Art. 127) base.s his demonstration of the theorem exclusive!}' upon 

 the priiiciiile of the conservation of areas, remarking at the outset, " if the earth and sea were so 

 entirely disconnected that one of them could revolve for any length of time with any velocity, in- 

 creasing or diminishing in any manner, while the other could revolve with any other velocity changing 

 in any other manner, we could pronounce nothing as to the effect of the fluctuation" (tidal) " upon 

 precession." 



A spheroidal nucleus wholly covered by an ocean of uniform depth, suffering no resistance, docs 

 not seem to me to lack much for fulfilling the above conditions. 



If velocities are generated in the waters of the ocean by solar (or lunar) attraction, the centrifugal 

 forces due to them might be looked to (though not alluded to by Lajilace) as agents for transferring, 

 from the fluid to the nucleus, the precession-producing couples due to the fluid mass, especially in 

 the above hypothetical case. It will be found, however, by reference to the expressions [2200], 

 that they give rise to no oouple, and are, moreover, very minute. 



The motion which the displacements [22G0] [22G1] indicate is a slight oscillation of the axis of 

 the fluid envelope, moving as a solid, about the axis of the nucleus, the angular distance between 

 these axes being slightly less than 2 seconds: it is, I presume, that which a non-rotating shell would 

 have were the attracting body, with constant distance and declination, to move, with angular velocity 

 M, in right ascension. In the case in hand it is the fluid shell which revolves and, suffering no 

 change of form, would be itself affected )jy its proper processional couple to the excluaiou of the 

 oscillation above described. 



ADDITIONAL NOTES. 



Note to page 11. 

 '" The process indicated is a more legitimate carrying out of the methods peculiar to this paper 

 than what follows in the text. The tangent of MM' (35) may be (approx.) taken for the sine, and the 

 cosine taken constant at unity, as may also be the cos /'. 



From (32) we may calculate by developing and neglecting terms in which sin' 7' enters 

 sin i = (1 — cos' i)< = sin / — sin I' cos / cos nj 

 sin i cos i = sin /cos I — sin /' cosSZcos nj — A sin'/' s\n2r cos'n^t 

 Introducing these values in (38) and (30), and integrating wo get expressions identical with 44 

 and 45, except a (practically) immaterial difference in the coefficient of I in the first which becomes 

 1 — ^ sin'/' instead of 1 — f sin'/'. 



Note to page 24. 



"' The foregoing interpretation of the symbolic integral in (T), adopted with hesitation from 

 authors cited, is based on assumed constancy of the angle <f. ; but this angle necessarily varies, 

 slowly indeed, but progressively, by the azimuthal motion measured by n sin x. The conditions for 

 the formation of a leminiscate are not, therefore, rigidly fulfilled. It will be found, however, taking 

 into account a complete excursion, that the slight increment which will enure to the moment of the 



quantity of motion, sin's -?., from this cause on one side of the vertical, will be neutralized on the 



at 



other, in consequence of the opposing signs of cos 4>, in opposite azimuths; or, at least, the rrsullaiit 



increment or decrement will ba a quantity of the second order in minuteness, and hence, affecting 



only in the same degree the azimuthal motion -~. 



dt 



Note to page 39. 

 '" The differential attraction of the sun on any length dx of the rod, at distance ;;; from the earth's 



/ S S \ 



centre is ( ^^ 1 dx, r being sun's distance. Integrate from x = x^o x^=R 'the earth's 



U'' — *) 1 ' 



radius). 



